Day 3: Logical Connectives and Compound Statements
Course Objectives
- Understand the meaning and usage of the five basic logical connectives
- Use logical connectives to construct compound statements
- Analyze the logical structure of conditional and biconditional statements
- Use truth tables to analyze the truth conditions of compound statements
Logical Connectives Basics
Logical connectives are words used to connect simple statements to form compound statements. In logic, there are five basic logical connectives:
- Conjunction (And, ∧): Expresses the relationship of "and"
- Disjunction (Or, ∨): Expresses the relationship of "or"
- Negation (Not, ¬): Expresses the relationship of "not"
- Conditional (If...then, →): Expresses the relationship of "if...then..."
- Biconditional (If and only if, ↔): Expresses the relationship of "if and only if"
Conjunction (And, ∧)
Conjunction is used to express the relationship of "and".
Symbol: ∧
Example: p ∧ q means "p and q"
Truth condition: p ∧ q is true only when p is true and q is true
Truth Table for Conjunction
| p | q | p ∧ q |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |
Disjunction (Or, ∨)
Disjunction is used to express the relationship of "or".
Symbol: ∨
Example: p ∨ q means "p or q"
Truth condition: p ∨ q is true when p is true or q is true (or both are true)
Truth Table for Disjunction
| p | q | p ∨ q |
|---|---|---|
| True | True | True |
| True | False | True |
| False | True | True |
| False | False | False |
Negation (Not, ¬)
Negation is used to express the relationship of "not".
Symbol: ¬
Example: ¬p means "not p" or "p is not the case"
Truth condition: The truth value of ¬p is the opposite of the truth value of p
Truth Table for Negation
| p | ¬p |
|---|---|
| True | False |
| False | True |
Conditional (If...then, →)
Conditional is used to express the relationship of "if...then...".
Symbol: →
Example: p → q means "if p, then q"
Truth condition: p → q is false only when p is true and q is false
Truth Table for Conditional
| p | q | p → q |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |
Biconditional (If and only if, ↔)
Biconditional is used to express the relationship of "if and only if".
Symbol: ↔
Example: p ↔ q means "p if and only if q"
Truth condition: p ↔ q is true when p and q have the same truth value
Truth Table for Biconditional
| p | q | p ↔ q |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | True |
Truth Tables
Truth tables are tools for analyzing the truth conditions of compound statements.
Steps for constructing a truth table:
- Identify the simple statements (atomic propositions) in the statement
- List all possible combinations of truth values
- Calculate the truth value of the compound statement based on the truth conditions of the logical connectives
Example of a Truth Table for a Compound Statement
Truth table for "(p ∧ q) → r":
| p | q | r | p ∧ q | (p ∧ q) → r |
|---|---|---|---|---|
| True | True | True | True | True |
| True | True | False | True | False |
| True | False | True | False | True |
| True | False | False | False | True |
| False | True | True | False | True |
| False | True | False | False | True |
| False | False | True | False | True |
| False | False | False | False | True |
Course Summary
In today's course, we learned:
- The five basic logical connectives: conjunction (AND, ∧), disjunction (OR, ∨), negation (NOT, ¬), conditional (IF...THEN, →), and biconditional (IF AND ONLY IF, ↔)
- The meaning and truth conditions of each logical connective
- How to use logical connectives to construct compound statements
- How to use truth tables to analyze the truth conditions of compound statements
Mind Map Overview
